Background

Summer \(T_s\) \[ \frac{dn}{d\tau} = n(1-n) - \frac{n^2p}{\tilde{b}^2+n^2}, \\ \frac{dp}{d\tau} = \tilde{\gamma} \frac{n^2p}{\tilde{b}^2+\tilde{n}^2} + \tilde{s}\frac{p}{1+\tilde{\nu}p}-\tilde{m}p \]

Winter \((1-T_s)\) \[ \frac{dn}{d\tau} = - \frac{\tilde{\alpha}np}{\tilde{\beta}+n}, \\ \frac{dp}{d\tau} = \tilde{\gamma} \frac{\tilde{\alpha}np}{\tilde{\beta}+n}-\tilde{\mu}p \]

Parameter Description
\(r\) Prey summer growth rate
\(K\) Prey carrying capacity
\(\alpha\) Specialist saturation killing rate
\(\beta\) Specialist halg-saturation
\(\gamma\) Predator-prey ration constant
\(\mu\) Winter predator death rate
\(s-m\) Predator intrinsic population growth
\(\nu\) Generalist density dependence

Research questions

Key results

Fluctuations in the system dynamics: “on-off” cycles

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Figure 1.1 - Wavelet spectrum for the model with a new ‘switching’ mechanism and no noise in the length of the summer season, T_s = 0.6
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Figure 1.2 - Time series
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Figure 1.3 - Poincare map
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Figure 2 - T_s = 0.6 + inc + noise(-0.05, 0.15) Wavelet spectrum of the time series (Fig. 4). Adding variability in summer length as well as a linear increase in summer length
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Figure 3 - Close-up examinations of the wavelet spectrum and the time series
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Figure 4 - T_s = 0.6 + inc + noise(-0.05,0.15) Simulated time series with increasing length and added variability in the summer season

Poincare maps for different season lengths

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Figure 5 - Poincare maps for varying period length, no noise in the length of summer

Limit cycles animations

Discussion and Future work


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References

1. Barraquand, F., Louca, S., Abbott, K. C., Cobbold, C. A., Cordoleani, F., DeAngelis, D. L., Elderd, B. D., Fox, J. W., Greenwood, P., Hilker, F. M., Murray, D. L., Stieha, C. R., Taylor, R. A., Vitense, K., Wolkowicz, G. S. K., & Tyson, R. C. (2017). Moving forward in circles: challenges and opportunities in modelling population cycles. Ecology Letters, 20(8), 1074–1092. https://doi.org/10.1111/ele.12789
2. Tyson, R., & Lutscher, F. (2016). Seasonally Varying Predation Behavior and Climate Shifts Are Predicted to Affect Predator-Prey Cycles. The American Naturalist, 188(5), 539–553. https://doi.org/10.1086/688665
3. Rinaldi, S., & Muratori, S. (1993). Conditioned chaos in seasonally perturbed predator-prey models. Ecological Modelling, 69(1-2), 79–97. https://doi.org/10.1016/0304-3800(93)90050-3
4. Torrence, C., & Compo, G. P. (1998). A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79(1), 61–78. https://doi.org/https://doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2